Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7 $
Sign $-0.174 - 0.984i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (1.11 + 1.32i)3-s + (0.499 + 0.866i)4-s − 5-s + (0.296 + 1.70i)6-s + (−0.142 − 2.64i)7-s + 0.999i·8-s + (−0.535 + 2.95i)9-s + (−0.866 − 0.5i)10-s + 4.86i·11-s + (−0.596 + 1.62i)12-s + (3.42 + 1.97i)13-s + (1.19 − 2.35i)14-s + (−1.11 − 1.32i)15-s + (−0.5 + 0.866i)16-s + (−1.25 + 2.16i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.640 + 0.767i)3-s + (0.249 + 0.433i)4-s − 0.447·5-s + (0.121 + 0.696i)6-s + (−0.0537 − 0.998i)7-s + 0.353i·8-s + (−0.178 + 0.983i)9-s + (−0.273 − 0.158i)10-s + 1.46i·11-s + (−0.172 + 0.469i)12-s + (0.948 + 0.547i)13-s + (0.320 − 0.630i)14-s + (−0.286 − 0.343i)15-s + (−0.125 + 0.216i)16-s + (−0.303 + 0.526i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.174 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.174 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.174 - 0.984i$
motivic weight  =  \(1\)
character  :  $\chi_{630} (311, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 630,\ (\ :1/2),\ -0.174 - 0.984i)\)
\(L(1)\)  \(\approx\)  \(1.50936 + 1.80095i\)
\(L(\frac12)\)  \(\approx\)  \(1.50936 + 1.80095i\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-1.11 - 1.32i)T \)
5 \( 1 + T \)
7 \( 1 + (0.142 + 2.64i)T \)
good11 \( 1 - 4.86iT - 11T^{2} \)
13 \( 1 + (-3.42 - 1.97i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.25 - 2.16i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.962 - 0.555i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 3.30iT - 23T^{2} \)
29 \( 1 + (-3.70 + 2.14i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.40 + 3.69i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.82 + 3.15i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.90 + 6.77i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.62 + 6.28i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.84 + 10.1i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.92 + 2.84i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.696 + 1.20i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.97 + 4.60i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.00 + 5.20i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.66iT - 71T^{2} \)
73 \( 1 + (-12.0 - 6.96i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.91 + 6.78i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.393 + 0.682i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.49 - 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-14.9 + 8.64i)T + (48.5 - 84.0i)T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−10.72051135661149419849365080625, −10.06443119509068725317552242115, −9.046036518172461836890104292854, −8.086755360301346491565794035068, −7.34533129134457254086078528538, −6.44493859792062706720014503459, −5.00265868920984207840504603792, −4.12719000256201453553353654698, −3.69462308562220527571616774545, −2.08266267775105387332022303230, 1.07234254807158215411778366456, 2.79419841022118076998175201484, 3.22534089175938400116702085046, 4.69130933069121948361761672255, 6.06314096108882553917117881630, 6.43271117023128235433652923058, 7.924851494029873008403646375475, 8.537255498061096014827229452095, 9.237731602217248444866115589726, 10.64808624372945347631445633924

Graph of the $Z$-function along the critical line